Answer
The given sets aren't equal for all sets $A$ and $B$.
Work Step by Step
Regions in $A'$ (all the regions not in $A$) are: $III, IV$
Regions in $B$ are: $II, III$
Regions in $A'\cap B$ are: $III$
Regions in $A$ are: $I, II$
Regions in $B'$ (all the regions not in $B$) are: $I, IV$
Regions in $A\cup B'$ are: $I, II, IV$
The elements of $A'\cap B$ and $A\cup B'$ are in different regions; therefore, those two sets aren't equal for all sets $A$ and $B$.
